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Hausdorff Convergence and Universal Covers \Lambda Christina Sormani Guofang Wei y

Summary: Hausdorff Convergence and Universal Covers \Lambda
Christina Sormani Guofang Wei y
We prove that if Y is the Gromov­Hausdorff limit of a sequence of compact manifolds, M n
i ,
with a uniform lower bound on Ricci curvature and a uniform upper bound on diameter, then
Y has a universal cover. We then show that, for i sufficiently large, the fundamental group of
M i has a surjective homeomorphism onto the group of deck transforms of Y . Finally, in the
non­collapsed case where the M i have an additional uniform lower bound on volume, we prove
that the kernels of these surjective maps are finite with a uniform bound on their cardinality.
A number of theorems are also proven concerning the limits of covering spaces and their deck
transforms when the M i are only assumed to be compact length spaces with a uniform upper
bound on diameter.
1 Introduction
In recent years the limit spaces of manifolds with lower bounds on Ricci curvature have been studied
from both a geometric and topological perspective. In particular, Cheeger and Colding have proven
a number of results regarding the regularity and geometric properties of these spaces. However,
the topology of the limit spaces is less well understood. Note that in this paper a manifold is a
Riemannian manifold without boundary.
Anderson [An] has proven that there are only finitely many isomorphism types of fundamental


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics